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人気のある三角関数 >

2+3cot^2(x)-3csc(x)=5

解

2 + 3 cot^{2} (x) - 3 csc (x) = 5

解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

+1

度

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

解答ステップ

2 + 3 cot^{2} (x) - 3 csc (x) = 5

両辺から

5

を引く

3 cot^{2} (x) - 3 csc (x) - 3 = 0

三角関数の公式を使用して書き換える

- 3 + 3 cot^{2} (x) - 3 csc (x)

ピタゴラスの公式を使用する:

1 + cot^{2} (x) = csc^{2} (x)

cot^{2} (x) = csc^{2} (x) - 1

= - 3 + 3 (csc^{2} (x) - 1) - 3 csc (x)

簡素化

- 3 + 3 (csc^{2} (x) - 1) - 3 csc (x) : 3 csc^{2} (x) - 3 csc (x) - 6

- 3 + 3 (csc^{2} (x) - 1) - 3 csc (x)

拡張

3 (csc^{2} (x) - 1) : 3 csc^{2} (x) - 3

3 (csc^{2} (x) - 1)

分配法則を適用する:

a (b - c) = ab - a c

a = 3, b = csc^{2} (x), c = 1

= 3 csc^{2} (x) - 3 \cdot 1

数を乗じる:

3 \cdot 1 = 3

= 3 csc^{2} (x) - 3

= - 3 + 3 csc^{2} (x) - 3 - 3 csc (x)

簡素化

- 3 + 3 csc^{2} (x) - 3 - 3 csc (x) : 3 csc^{2} (x) - 3 csc (x) - 6

- 3 + 3 csc^{2} (x) - 3 - 3 csc (x)

条件のようなグループ

= 3 csc^{2} (x) - 3 csc (x) - 3 - 3

数を引く:

- 3 - 3 = - 6

= 3 csc^{2} (x) - 3 csc (x) - 6

= 3 csc^{2} (x) - 3 csc (x) - 6

= 3 csc^{2} (x) - 3 csc (x) - 6

- 6 - 3 csc (x) + 3 csc^{2} (x) = 0

置換で解く

- 6 - 3 csc (x) + 3 csc^{2} (x) = 0

仮定:

csc (x) = u

- 6 - 3 u + 3 u^{2} = 0

- 6 - 3 u + 3 u^{2} = 0 : u = 2, u = - 1

- 6 - 3 u + 3 u^{2} = 0

標準的な形式で書く

a x^{2} + b x + c = 0

3 u^{2} - 3 u - 6 = 0

解くとthe二次式

3 u^{2} - 3 u - 6 = 0

二次Equationの公式:

次の場合:

a = 3, b = - 3, c = - 6

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 3 ( - 6 )}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 3 ( - 6 )}{2 \cdot 3}

(- 3)^{2} - 4 \cdot 3 (- 6) = 9

(- 3)^{2} - 4 \cdot 3 (- 6)

規則を適用

- (- a) = a

= (- 3)^{2} + 4 \cdot 3 \cdot 6

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

(- 3)^{2} = 3^{2}

= 3^{2} + 4 \cdot 3 \cdot 6

数を乗じる:

4 \cdot 3 \cdot 6 = 72

= 3^{2} + 72

3^{2} = 9

= 9 + 72

数を足す:

9 + 72 = 81

= 81

数を因数に分解する:

81 = 9^{2}

= 9^{2}

累乗根の規則を適用する:

n a^{n} = a

9^{2} = 9

= 9

u_{1, 2} = \frac{- ( - 3 ) \pm 9}{2 \cdot 3}

解を分離する

u_{1} = \frac{- ( - 3 ) + 9}{2 \cdot 3}, u_{2} = \frac{- ( - 3 ) - 9}{2 \cdot 3}

u = \frac{- ( - 3 ) + 9}{2 \cdot 3} : 2

\frac{- ( - 3 ) + 9}{2 \cdot 3}

規則を適用

- (- a) = a

= \frac{3 + 9}{2 \cdot 3}

数を足す:

3 + 9 = 12

= \frac{12}{2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{12}{6}

数を割る:

\frac{12}{6} = 2

= 2

u = \frac{- ( - 3 ) - 9}{2 \cdot 3} : - 1

\frac{- ( - 3 ) - 9}{2 \cdot 3}

規則を適用

- (- a) = a

= \frac{3 - 9}{2 \cdot 3}

数を引く:

3 - 9 = - 6

= \frac{- 6}{2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{- 6}{6}

分数の規則を適用する:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6}{6}

規則を適用

\frac{a}{a} = 1

= - 1

二次equationの解:

u = 2, u = - 1

代用を戻す

u = csc (x)

csc (x) = 2, csc (x) = - 1

csc (x) = 2, csc (x) = - 1

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = 2

以下の一般解

csc (x) = 2

csc (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = - 1 : x = \frac{3 π}{2} + 2 πn

csc (x) = - 1

以下の一般解

csc (x) = - 1

csc (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

すべての解を組み合わせる

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

グラフ

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